Séminaire Lotharingien de Combinatoire, 95B.7 (2026), 12 pp.
Mohammad Bardestani, Keivan Mallahi-Karai, Samrith Ram and Hadi Salmasian
Point Counting on Intersections of Adjoint Orbits with
Ad-Nilpotent Ideals over Fq
Abstract.
Let bn(Fq) denote
the Lie algebra of upper triangular n × n matrices
with entries in the finite field Fq. For
every
bn(Fq)-stable ideal
a of the Lie
algebra un(Fq) of
strictly upper triangular matrices, and every partition μ
of n, we obtain an explicit formula for the number of elements
of a of Jordan type μ. Up to an explicit factor depending
on q,
our formula is given by the Hall scalar product of a modified
Hall-Littlewood function indexed by μ and a chromatic quasisymmetric function associated to a.
In the special case that a
= uΛ(Fq), where
uΛ(Fq) is the
nilradical of the standard parabolic subalgebra
of gln(Fq)
associated to a composition Λ of n, our formula reduces
to the statement (recently proved by Karp and Thomas) that
up to an explicit polynomial factor in q, the number of
elements in
uΛ(Fq) of Jordan
type μ is equal to the coefficient of the monomial
xΛ in the specialization of the dual
Macdonald symmetric function
Qμ'(x;q-1,t) at t=0.
We obtain a new and shorter proof of the latter assertion
using a parabolic variation of Borodin's division algorithm.%~\cite{Borodin,Kirillov}.
We give four applications of our main result stated above.
First, we obtain an analogous formula for the number of points of a nilpotent Hessenberg variety.
Second, we demonstrate that a recurrence relation of Kirillov %in~\cite{KirillovX=0}
for the number of strictly upper triangular matrices of Jordan type
μ is a straightforward consequence of our main result combined with
the Cauchy identity for Macdonald polynomials. Third, we give an
explicit formula for the number of X
∈ uΛ(Fq) that
satisfy X2=0. In the special case Λ =
(1n), our formula is different from the one
conjectured by Kirillov and Melnikov
(and proved by Ekhad and Zeilberger). However, we obtain a new proof of the Kirillov-Melnikov-Ekhad-Zeilberger
formula
using our latter result and a formula for two-rowed Macdonald polynomials by Jing and Józefiak.
Fourth, we give an explicit formula for the number of double cosets
U1\GLn(Fq)/U2
where U1 and U2 denote unipotent subgroups of
GLn(Fq) that correspond to
two bn(Fq)-stable
ideals of un(Fq) .
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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